Answer
The function for this function is y=(${{\frac{2}{3})}^x}$.
The graph for this function is in the screenshot below.
Work Step by Step
The function for this function is y=(${{\frac{2}{3})}^x}$ because the y coordinates are increasing exponentially and not in a linear relationship. The function was created by finding the value of b using the equation of y=$b^{x}$ which is used for exponential function. The value of b was founded by dividing the y values of the subsequent x values.
(1, $\frac{2}{3}$), (2, $\frac{4}{9}$), (3, $\frac{8}{27}$), (4, $\frac{16}{81}$), (5, $\frac{32}{243}$)
b value of $x_{2}$ - $x_{1}$ = $\frac{\frac{4}{9}}{\frac{2}{3}}$ = $\frac{4}{9}$ $\times$ $\frac{3}{2}$ = $\frac{2}{3}$
b value of $x_{3}$ - $x_{2}$ = $\frac{\frac{8}{27}}{\frac{4}{9}}$ = $\frac{8}{27}$ $\times$ $\frac{9}{4}$ = $\frac{2}{3}$
b value of $x_{4}$ - $x_{3}$ = $\frac{\frac{16}{81}}{\frac{8}{27}}$ = $\frac{16}{81}$ $\times$ $\frac{27}{8}$ = $\frac{2}{3}$
b value of $x_{5}$ - $x_{4}$ = $\frac{\frac{32}{243}}{\frac{16}{81}}$ = $\frac{32}{243}$ $\times$ $\frac{81}{16}$ = $\frac{2}{3}$
Therefore the value of b is $\frac{2}{3}$ and thus the function is y=(${{\frac{2}{3})}^x}$
